In this paper, some special distance spectral properties of graphs are considered. Concretely, we recursively construct an infinite family trees with eigenvalue −1, and determine all triangle-free connected three distinct eigenvalues which the smallest one is equal to −3. Furthermore, characterize eigenvalues.

In 1986, Terwilliger showed that there is a strong relation between the eigenvalues of a distance-regular graph and the eigenvalues of a local graph. In particular, he showed that the eigenvalues of a local graph are bounded in terms of the eigenvalues of a distance-regular graph, and he also showed that if an eigenvalue θ of the distance-regular graph has multiplicity m less than its valency k...

Let the eigenvalues of a graph and the angles between eigenspaces and the co-ordinate axes of the corresponding real vector space be given for a graph. Cvetković [2] gave a method of constructing a graph which is the supergraph of all graphs with given eigenvalues and angles. Based on this, we describe a branch & bound algorithm for constructing all graphs with given eigenvalues and angles.

bandwidth labelling is a well known research area in graph theory. we provide a new proof that the bandwidth of mobius ladder is 4, if it is not a $k_{4}$, and investigate the bandwidth of a wider class of mobius graphs of even strips.